Why is regret so defined in MABs?
In short, you don't regret your bad luck that you could do nothing about, you regret your bad choices that you could have done something about if only you knew.
The point of regret as a metric therefore is to compare your choices with the ideal choices. This makes sense in MABs, because although the primary goal is to gain the most reward, the learning part of the goal is to calculate from experience what are the best choices - usually whilst sacrificing as little as possible in the process.
The formula captures that concept, so does not concern itself with individual rewards in the past that could have been due to good or bad luck. Hence it uses expected (or mean) rewards.